Department of

Mathematics

Seminar Calendar
for events the day of Tuesday, October 20, 2020.

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Questions regarding events or the calendar should be directed to Tori Corkery.
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Tuesday, October 20, 2020

11:00 am in via Zoom,Tuesday, October 20, 2020

Scarcity of congruences for the partition function

Scott Ahlgren (UIUC Math)

Abstract: The arithmetic properties of the ordinary partition function p(n) have been the topic of intensive study. Ramanujan proved that there are linear congruences of the form p(ℓn+β)≡0(modℓ) for the primes ℓ=5,7,11, and it is known that there are no others of this form. On the other hand, there are many examples of congruences of the form p(ℓQmn+β)≡0(modℓ) where Q is prime and m≥3. Here we prove that such congruences are very scarce in the case when m=1 or m=2. The proofs rely on a variety of tools from the theory of modular forms and from analytic number theory. This is joint work with Olivia Beckwith and Martin Raum. Please contact Kevin Ford (ford@math.uiuc.edu) for Zoom link.

11:00 am in via Zoom,Tuesday, October 20, 2020

The Picard group of the stable module category for quaternion groups

Richard Wong (University of Texas at Austin)

Abstract: One problem of interest in modular representation theory of finite groups is in computing the group of endo-trivial modules. In homotopy theory, this group is known as the Picard group of the stable module category. This group was originally computed by Carlson-Thévenaz using the theory of support varieties. However, I provide new, homotopical proofs of their results for the quaternion group of order 8, and for generalized quaternion groups, using the descent ideas and techniques of Mathew and Mathew-Stojanoska. Notably, these computations provide conceptual insight into the classical work of Carlson-Thévenaz. (Please email vesna@illinois.edu for Zoom link.)

1:00 pm in Zoom,Tuesday, October 20, 2020

Ordered paths and the Erdős–Szekeres theorem

Misha Lavrov (Kennesaw State University)

Abstract: This talk is about several Ramsey-type problems on ordered graphs motivated by the Erdos–Szekeres theorem. This theorem guarantees that any sequence of $rs+1$ distinct numbers has an increasing subsequence of length $r+1$ or a decreasing subsequence of length $s+1$. In another, stronger formulation: any red-blue coloring of an ordered complete graph on $rs+1$ vertices, there is a red ordered path of length $r$ or a blue ordered path of length $s$.

Our main result characterizes all ordered graphs on $rs+1$ vertices for which the same conclusion holds. We also apply our techniques to prove bounds on the online coloring number (for any number of colors). In the monotone subsequence formulation, the online problem is a natural algorithmic question: how many comparisons are necessary to find a monotone subsequence guaranteed by the Erdos–Szekeres theorem?

These results are joint work with Jozsef Balogh, Felix Clemen, and Emily Heath.